Order functions of plurisubharmonic functions

Halil Celik; Evgeny Poletsky

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Mordechay B. Levin (2001)

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Let $q, q_{1}, \dots, q_{s} \geq 2$ be integers, and let $α_{1}, α_{2}, \dots$ be a sequence of real numbers. In this paper we prove that the lower bound of the discrepancy of the double sequence ${(\{α_{m} q^{n}\}, \dots, \{α_{m + s - 1} q^{n}\})}_{m, n = 1}^{M N}$ coincides (up to a logarithmic factor) with the lower bound of the discrepancy of ordinary sequences ${(x n)}_{n = 1}^{M N}$ in $s$ -dimensional unit cube $(s, M, N = 1, 2, \dots)$ . We also find a lower bound of the discrepancy (up to a logarithmic factor) of the sequence ${(\{α_{1} q_{1}^{n}\}, \dots, \{α_{s} q_{s}^{n}\})}_{n = 1}^{N}$ (Korobov’s problem).

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Let $r (n, m)$ denote the number of partitions of $n$ into parts, each of which is at least $m$ . By applying the saddle point method to the generating series, an asymptotic estimate is given for $r (n, m)$ , which holds for $n \to \infty$ , and $1 \leq m \leq c_{1} \frac{n}{{(log n)}^{c_{2}}}$ .

Weighted integrability and L¹-convergence of multiple trigonometric series

Chang-Pao Chen (1994)

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We prove that if $c_{j k} \to 0$ as max(|j|,|k|) → ∞, and $\sum_{| j | = 0 \pm}^{\infty} \sum_{| k | = 0 \pm}^{\infty} {θ (| j |}^{⊤} {) ϑ (| k |}^{⊤}) | Δ_{12} c_{j k} | < \infty$ , then f(x,y)ϕ(x)ψ(y) ∈ L¹(T²) and $\iint_{T ²} | s_{m n} (x, y) - f (x, y) | \cdot | ϕ (x) ψ (y) | d x d y \to 0$ as min(m,n) → ∞, where f(x,y) is the limiting function of the rectangular partial sums $s_{m n} (x, y)$ , (ϕ,θ) and (ψ,ϑ) are pairs of type I. A generalization of this result concerning L¹-convergence is also established. Extensions of these results to double series of orthogonal functions are also considered. These results can be extended to n-dimensional case. The aforementioned results generalize work of Balashov [1],...